The next natural step from the stereo formalism and the fundamental
matrix is a multi-camera situation (i.e. 3 or more projections). The
trifocal tensor approach is such an extension and maintains a similar
projective geometry spirit. This model has been proposed and developed by Sashua
[46], Hartley [23] and
Faugeras [19] among others.
Figure 5 represents the imaging scenario.

Figure:
Trifocal Geometry

Here, the trifocal plane is formed by the three optic centers COP1,
COP2 and COP3. Intersecting this plane with the three image
planes produces three lines called the trifocal lines t1, t2 and
t3. There are now two epipoles (the ei,j). One could use
standard epipolar geometry and consider three fundamental matrices
(one for each pair of COPs) F12, F23 and F31. However,
the fundamental matrices are subject to some standard limitations
which might be avoidable here. For instance, if a point P is in the
trifocal plane, the fundamental matrices cannot determine if its 3
images belong to a single 3D point. In fact, there is additional
information in the three plane case. Given a point in one image, it is
possible to construct a line in another using the fundamental
matrix. However, given a point in the first image and a point in the
second image, one can directly compute the coordinates of the third
point using a structure called the trifocal tensor which is the
analog of the fundamental matrix for 3 view situations. Typically, one
uses this tensor (denoted )
to map a line in image 1 (l1)
and a line in image 2 (l2) to a line in image 3 (l3). This
mapping is again a linear expression as in
Equation 5.

(5)

To map points, one merely considers intersections of mapped lines. The
tensor
can be considered as a
3 x 3 x 3 cube
operator (i.e. defined by 27 scalars in total). It can also be
represented as the concatenation of three 3 x 3
matrices: G1, G2 and G3 which allow us to expand the above
into the more straightforward Equation 6.

(6)

If a set of corresponded points are known in each of the 3 images,
the tensor can be estimated in a similar way as the fundamental
matrix. For instance, one can perform a least-squares linear
computation to recover the 27 parameters [23]
[46]. However, the trifocal tensor's 27 scalar
parameters are not all independent unknowns. Not every
3 x 3
x 3 cube is a tensor. It too has constraints (like the
fundamental matrix) and really has only 18 degrees of freedom. The
above linear methods for recovering the tensor do not impose the
constraints and can therefore produce invalid tensors.

By making an appeal to Grassmann-Cayley algebra, Faugeras gracefully
derives the algebraic constraints on trifocal tensors which can be
viewed as higher order (4th degree) polynomials on the parameters
[19]. The 9 constraints are folded into a
nonlinear optimization scheme which recovers the 18 remaining degrees
of freedom of the tensor from image correspondences.